chap7-rev-problems.xml

<?xml version="1.0"?>


<!-- This file was originally part of the book     -->
<!--   Modeling, Functions, and Graphs   -->
<!--           4th                                     -->
<!-- Copyright (C) Katherine Yoshiwara      -->

<exercises xml:id="chap7-rev-problems"  xmlns:xi="http://www.w3.org/2001/XInclude">
    <title>Review Problems</title>

<exercisegroup cols="2"><introduction><p>Multiply.</p></introduction>

<exercise number="1">
    <statement><p><m>(2x - 5)(x^2 - 3x + 2)</m></p></statement>
    <answer><p><m>2x^3 - 11x^2 + 19x - 10 </m></p></answer>
</exercise>

<exercise number="2"><statement><p><m>(b^2 - 2b - 3)(2b^2 + b - 5) </m></p></statement></exercise>

<exercise number="3"><statement><p><m>(t + 4)(t^2 - t - 1)</m></p></statement>
<answer><p><m>t^3 + 3t^2 - 5t - 4</m></p></answer>
</exercise>

<exercise number="4"><statement><p><m>(b + 3)(2b - 1)(2b + 5) </m></p></statement></exercise>
</exercisegroup>



<exercisegroup ><introduction><p>Find the indicated term.</p></introduction>

<exercise number="5"><statement><p><m>(1 - 3x + 5x^2)(7 + x - x^2)\text{;} ~~x^2</m></p></statement>
<answer><p><m>31x^2</m></p></answer>
</exercise>

<exercise number="6"><statement><p><m>(-3 + x - 4x^2)(4 + 3x - 2x^3)\text{;} ~~x^3</m></p></statement>
</exercise>

<exercise number="7"><statement><p><m>(4x - x^2 + 3x^3)(1 + 4x - 3x^2)\text{;} ~~x^3</m></p></statement>
<answer><p><m>-13x^3</m></p></answer>
</exercise>

<exercise number="8"><statement><p><m>(3 - 2x + 2x^3)(5 + 3x - 2x^2 + 4x^4)\text{;} ~~x^4</m></p></statement>
</exercise>
</exercisegroup>


<exercisegroup cols="2"><introduction><p>Factor.</p></introduction>

<exercise number="9"><statement><p><m>8x^3-27z^3 </m></p></statement>
<answer><p><m>(2x-3z)(4x^2+6xz+9z^2) </m></p></answer>
</exercise>

<exercise number="10"><statement><p><m>1 + 125a^3b^3 </m></p></statement>
</exercise>

<exercise number="11"><statement><p><m>y^3 + 27x^3 </m></p></statement>
<answer><p><m>(y + 3x)(y^2 - 3xy + 9x^2) </m></p></answer>
</exercise>

<exercise number="12"><statement><p><m>x^9 - 8 </m></p></statement>
</exercise>
</exercisegroup>


<exercisegroup cols="2"><introduction><p>Write as a polynomial.</p></introduction>

<exercise number="13"><statement><p><m>(v - 10)^3</m></p></statement>
<answer><p><m>v^3 - 30v^2 + 300v - 1000</m></p></answer>
</exercise>

<exercise number="14"><statement><p><m>(a + 2b^2)^3</m></p></statement></exercise>
</exercisegroup>


<exercise number="15"><statement><p>
    The expression <m>\dfrac{n}{6 }(n - 1)(n - 2)</m> gives the number of different <m>3</m>-item pizzas that can be created from a list of <m>n</m> toppings.
    <ol>
        <li><p>Write the expression as a polynomial.</p></li>
        <li><p>If Mitch's Pizza offers <m>12</m> different toppings, how many different combinations for <m>3</m>-item pizzas can be made?</p></li>
        <li><p>Use a table or graph to determine how many different toppings are needed in order to be able to have more than <m>1000</m> possible combinations for <m>3</m>-item pizzas.</p></li>
    </ol>
</p></statement>
<answer><p><ol>
    <li><p><m>\dfrac{1}{6}n^3-\dfrac{1}{2}n^2+\dfrac{1}{3}n </m></p></li>
    <li><p><m>220</m></p></li>
    <li><p><m>20</m></p></li>
</ol></p></answer>
</exercise>

<exercise number="16"><statement><p>
    The expression <m>n(n - 1)(n - 2)</m> gives the number of different triple-scoop ice cream cones that can be created from a list of <m>n</m> flavors.
    <ol>
        <li><p>Write the expression as a polynomial.</p></li>
        <li><p>If Zanner's Ice Cream Parlor offers <m>21</m> flavors, how many different triple-scoop ice cream cones can be made?</p></li>
        <li><p>Use a table or graph to determine how many different flavors are needed in order to be able to have more than <m>10,000</m> possible triple-scoop ice cream cones.</p></li>
    </ol>
</p></statement>
</exercise>


<exercisegroup><introduction><p>
    For Problems 17 and 18,
    <ol label="a">
        <li><p>Graph each polynomial in the standard window.</p></li>
        <li><p>Find the range of the function on the domain <m>[-10, 10]</m>.</p></li>
    </ol>
</p></introduction>

<exercise number="17"><statement><p><m>f (x) = x^3 - 3x + 2</m></p></statement>
<answer><p><ol cols="2">
    <li><p><image source="images/fig-ans-chap7-rev-17.jpg"   width="80%"><description>cubic</description></image> </p></li>
    <li><p><m>[-968, 972]</m></p></li>
</ol></p></answer>
</exercise>

<exercise number="18"><statement><p><m>g(x) = -0.1(x^4 - 6x^3 + x^2 + 24x + 16)</m></p></statement></exercise>
</exercisegroup>


<exercisegroup cols="2"><introduction><p>
    For Problems 19<ndash/>28,
    <ol label="a">
        <li><p>Find the zeros of the polynomial.</p></li>
        <li><p>Sketch the graph by hand.</p></li>
    </ol>
</p></introduction>

<exercise number="19"><statement><p><m>f (x) = (x - 2)(x + 1)^2</m></p></statement>
<answer><p><ol cols="2">
    <li><p><m>2, -1</m></p></li>
    <li><p><image source="images/fig-ans-chap7-rev-19"  width="80%"><description>cubic</description></image> </p></li>
</ol></p></answer>
</exercise>

<exercise number="20"><statement><p><m>g(x) = (x - 3)^2(x + 2)</m></p></statement>
</exercise>

<exercise number="21"><statement><p><m>G(x) = x^2(x - 1)(x + 3)</m></p></statement>
<answer><p><ol cols="2">
    <li><p><m>0,1, -3</m></p></li>
    <li><p><image source="images/fig-ans-chap7-rev-21"  width="80%"><description>quartic</description></image> </p></li>
</ol></p></answer>
</exercise>

<exercise number="22"><statement><p><m>F(x) = (x + 1)^2(x - 2)^2</m></p></statement>
</exercise>

<exercise number="23"><statement><p><m>V(x) = x^3 - x^5</m></p></statement>
<answer><p><ol cols="2">
    <li><p><m>0,1, -1</m></p></li>
    <li><p><image source="images/fig-ans-chap7-rev-23"  width="80%"><description>quintic</description></image> </p></li>
</ol></p></answer>
</exercise>

<exercise number="24"><statement><p><m>H(x) = x^4 - 9x^2</m></p></statement>
</exercise>

<exercise number="25"><statement><p><m>P(x) = x^3 + x^2 - x - 1</m></p></statement>
<answer><p><ol cols="2">
    <li><p><m>-1, 1</m></p></li>
    <li><p><image source="images/fig-ans-chap7-rev-25"  width="80%"><description>cubic</description></image> </p></li>
</ol></p></answer>
</exercise>

<exercise number="26"><statement><p><m>y = x^3 + x^2 - 2x</m></p></statement>
</exercise>

<exercise number="27"><statement><p><m>y = x^4 - 7x^2 + 6</m></p></statement>
<answer><p><ol cols="2">
    <li><p><m>-1, 1, \pm\sqrt{6} </m></p></li>
    <li><p><image source="images/fig-ans-chap7-rev-27"  width="80%"><description>quartic</description></image> </p></li>
</ol></p></answer>
</exercise>

<exercise number="28"><statement><p><m>y = x^4 + x^3 - 3x^2 - 3x</m></p></statement>
</exercise>
</exercisegroup>


<exercisegroup cols="2"><introduction><p>Find a possible formula for the polynomial, in factored form.</p></introduction>

<exercise number="29"><statement><p><sidebyside width="60%"><image source="images/fig-chap7-rev-29"><description>cubic</description></image> </sidebyside></p></statement>
<answer><p><m>x(x + 2)(x - 3)</m></p></answer>
</exercise>

<exercise number="30"><statement><p><sidebyside width="60%"><image source="images/fig-chap7-rev-30"><description>cubic</description></image> </sidebyside></p></statement>
</exercise>

<exercise number="31"><statement><p><sidebyside width="60%"><image source="images/fig-chap7-rev-31"><description>quintic</description></image> </sidebyside></p></statement>
<answer><p><m>x^3(x + 2)(x - 2)</m></p></answer>
</exercise>

<exercise number="32"><statement><p><sidebyside width="60%"><image source="images/fig-chap7-rev-32"><description>quintic</description></image> </sidebyside></p></statement>
</exercise>

<exercise number="33"><statement><p><sidebyside width="60%"><image source="images/fig-chap7-rev-33"><description>quartic</description></image> </sidebyside></p></statement>
<answer><p><m>x^2(x + 4)(x - 4)</m></p></answer>
</exercise>

<exercise number="34"><statement><p><sidebyside width="60%"><image source="images/fig-chap7-rev-32"><description>quartic</description></image> </sidebyside></p></statement>
</exercise>
</exercisegroup>



<exercisegroup ><introduction><p>
    For Problems 35<ndash/>36,
    <ol label="a">
        <li><p>Verify that the given value is a zero of the polynomial.</p></li>
        <li><p>Find the other zeros. (<em>Hint</em>: Use polynomial division to write <m>P(x) = (x - a)Q(x)</m>, then factor <m>Q(x)</m>.)</p></li>
    </ol>
</p></introduction>

<exercise number="35"><statement><p><m>P(x) = x^3 - x^2 - 7x - 2</m>; <m>a=-2</m></p></statement>
<answer><p><ol cols="2">
    <li><p><m>P(-2) = 0</m></p></li>
    <li><p><m>\dfrac{3\pm\sqrt{13}}{2} </m></p></li>
</ol></p></answer>
</exercise>

<exercise number="36"><statement><p><m>P(x) = 3x^3 - 11x^2 - 5x + 4</m>; <m>a=4</m></p></statement>
</exercise>
</exercisegroup>


<exercisegroup cols="2"><introduction><p>
    For Problems 37<ndash/>40,
    <ol label="a">
        <li><p>Solve the quadratic equation, and write the solutions in the form <m>a + bi</m>.</p></li>
        <li><p>Check your solutions.</p></li>
    </ol>
</p></introduction>

<exercise number="37"><statement><p><m>x^2 + 4x + 10 = 0</m></p></statement>
<answer><p><m>-2 \pm i \sqrt{6}</m></p></answer>
</exercise>

<exercise number="38"><statement><p><m>x^2 - 2x + 7 = 0</m></p></statement></exercise>

<exercise number="39"><statement><p><m>3x^2 - 6x + 5 = 0</m></p></statement>
<answer><p><m>1 \pm \dfrac{\sqrt{6}}{3} i </m></p></answer>
</exercise>

<exercise number="40"><statement><p><m>2x^2 + 5x + 4 = 0</m></p></statement></exercise>
</exercisegroup>


<exercisegroup cols="2"><introduction><p>Evaluate each polynomial for the given value of the variable.</p></introduction>

<exercise number="41"><statement><p>
    <m>z^2 - 6z + 5</m>
    <ol >
        <li><p><m>z = 3 + 2i</m></p></li>
        <li><p><m>z=3-2i</m></p></li>
    </ol>
</p></statement>
<answer><p><ol cols="2">
    <li><p><m>-8</m></p></li>
    <li><p><m>-8</m></p></li>
</ol></p></answer>
</exercise>

<exercise number="42"><statement><p>
    <m>w^2 + 4w + 7</m>
    <ol >
        <li><p><m>w= -1 - 3i</m></p></li>
        <li><p><m>w = -1 + 3i</m></p></li>
    </ol>
</p></statement></exercise>
</exercisegroup>


<exercisegroup cols="2"><introduction><p>Find the quotient.</p></introduction>

<exercise number="43"><statement><p><m>\dfrac{2-5i}{3-i} </m></p></statement>
<answer><p><m>\dfrac{11}{10}-\dfrac{13}{10}i </m></p></answer>
</exercise>

<exercise number="44"><statement><p><m>\dfrac{1+i}{1-i} </m></p></statement>
</exercise>
</exercisegroup>


<exercisegroup cols="2"><introduction><p>Find a fourth-degree polynomial with the given zeros.</p></introduction>

<exercise number="45"><statement><p><m>3i, 1-2i</m></p></statement>
<answer><p><m>x^4 - 2x^3 + 14x^2 - 18x + 45</m></p></answer>
</exercise>

<exercise number="46"><statement><p><m>2-\sqrt{3} i, 2+3i</m></p></statement></exercise>
</exercisegroup>


<exercisegroup cols="2"><introduction><p>Plot each complex number as a point on the complex plane.</p></introduction>

<exercise number="47"><statement><p><ol>
    <li><p><m>z = 4 + i, ~\overline{z}, ~-z, ~-\overline{z}</m></p></li>
    <li><p><m>iz, ~i\overline{z}, ~-iz, ~-i\overline{z}</m></p></li>
</ol></p></statement>
<answer><p><ol cols="2">
    <li><p><image source="images/fig-ans-chap7-rev-47"  width="90%"><description>four points in complex plane</description></image> </p></li>
    <li><p><image source="images/fig-ans-chap7-rev-47b"  width="90%"><description>four points in complex plane</description></image> </p></li>
</ol></p></answer>
</exercise>

<exercise number="48"><statement><p><ol>
    <li><p><m>w = -2 + 3i, ~\overline{w}, ~-w, ~-\overline{w}</m></p></li>
    <li><p><m>iw, ~i\overline{w}, ~-iw, ~-i\overline{w}</m></p></li>
</ol></p></statement>
</exercise>
</exercisegroup>


<exercise number="49"><statement><p>
    The radius, <m>r</m>, of a cylindrical can should be one-half its height, <m>h</m>.
    <ol>
        <li><p>Express the volume, <m>V</m>, of the can as a function of its height.</p></li>
        <li><p>What is the volume of the can if its height is <m>2</m> centimeters? <m>4</m> centimeters?</p></li>
        <li><p>Graph the volume as a function of the height and verify your results of part (b) graphically. What is the approximate height of the can if its volume is <m>100</m> cubic centimeters?</p></li>
    </ol>
</p></statement>
<answer><p><ol>
    <li><p><m>V=\dfrac{\pi h^3}{4} </m></p></li>
    <li><p><m>2\pi\text{ cm}^3 \approx 6.28\text{ cm}^3 </m>; <m>16\pi\text{ cm}^3 \approx 50.27\text{ cm}^3 </m> </p></li>
    <li><p><image source="images/fig-ans-chap7-rev-49.jpg"  width="30%"><description>cubic</description></image> </p></li>
</ol></p></answer>
</exercise>

<exercise number="50"><statement><p>
    The Twisty-Freez machine dispenses soft ice cream in a cone-shaped peak with a height <m>3</m> times the radius of its base. The ice cream comes in a round bowl with base diameter <m>d</m>.
    <ol>
        <li><p>Express the volume, <m>V</m>, of  Twisty-Freez in the bowl as a function of <m>d</m>.</p></li>
        <li><p>How much Twisty-Freez comes in a <m>3</m>-inch diameter dish? A <m>4</m>-inch dish?</p></li>
        <li><p>Graph the volume as a function of the diameter and verify your results of part (b) graphically. What is the approximate diameter of a Twisty-Freez if its volume is <m>5</m> cubic inches?</p></li>
    </ol>
</p></statement>
</exercise>

<exercise><statement><p>
    A new health club opened up, and the manager kept track of the number of active members over its first few months of operation. The equation below gives the number, <m>N</m>, of active members, in hundreds, <m>t</m> months after the club opened.
    <me>N=\frac{44t}{40+t^2}</me>
    <ol>
        <li><p>Use your calculator to graph the function <m>N</m> on a suitable domain.</p></li>
        <li><p>How many active members did the club have after <m>8</m> months?</p></li>
        <li><p>In which months did the club have <m>200</m> active members?</p></li>
        <li><p>When does the health club have the largest number of active members? What happens to the number of active members as time goes on?</p></li>
    </ol> 
</p></statement>
<answer><p><ol>
    <li><p><image source="images/fig-ans-chap7-rev-51.jpg"  width="30%"><description>rational function</description></image> </p></li>
    <li><p><m>338</m></p></li>
    <li><p>Months <m>2</m> and <m>20</m></p></li>
    <li><p>During month <m>6</m>. The number of members eventually decreases to zero.</p></li>
</ol></p></answer>
</exercise>

<exercise number="52"><statement><p>
    A small lake in a state park has become polluted by runoff from a factory upstream. The cost for removing <m>p</m> percent of the pollution from the lake is given, in thousands of dollars, by
    <me>C=\frac{25p}{100-p} </me>
    <ol>
        <li><p>Use your calculator to graph the function <m>C</m> on a suitable domain.</p></li>
        <li><p>How much will it cost to remove <m>40\%</m> of the pollution?</p></li>
        <li><p>How much of the pollution can be removed for <m>\$100,000</m>?</p></li>
        <li><p>What happens to the cost as the amount of pollution to be removed increases? How much will it cost to remove all the pollution?</p></li>
    </ol>
</p></statement></exercise>


<exercisegroup cols="2"><introduction><p>State the domain of each function.</p></introduction>

<exercise number="53"><statement><p><m>h(x)=\dfrac{x^2-9}{x(x^2-4)} </m></p></statement>
<answer><p>All numbers except <m>-2, 0, 2</m>.</p></answer>
</exercise>

<exercise number="54"><statement><p><m>f(x)=\dfrac{x^2-3x+10}{x^2(x^2+1)} </m></p></statement></exercise>
</exercisegroup>


<exercisegroup cols="2"><introduction><p>
    For Problems 55<ndash/>56,
    <ol label="a">
        <li><p>Sketch the horizontal and vertical asymptotes for each function.</p></li>
        <li><p>Use the asymptotes to help you sketch the graph.</p></li>
    </ol>
</p></introduction>

<exercise number="55"><statement><p><m>F(x)=\dfrac{2x}{x^2-1} </m></p></statement>
<answer><p><image source="images/fig-ans-chap7-rev-55"  width="45%"><description>rational function</description></image> </p></answer>
</exercise>

<exercise number="56"><statement><m>G(x)=\dfrac{2}{x^2-1} </m></statement></exercise>
</exercisegroup>


<exercisegroup cols="3"><introduction><p>
    For Problems 57<ndash/>62,
    <ol>
        <li><p>Identify all asymptotes and intercepts.</p></li>
        <li><p>Sketch the graph.</p></li>
    </ol>
</p></introduction>

<exercise number="57"><statement><p><m>y=\dfrac{1}{x-4} </m></p></statement>
<answer><p><ol>
    <li><p>Horizontal asymptote <m>y = 0</m>; Vertical asymptote <m>x = 4</m>; <m>y</m>-intercept <m>(0,\frac{-1}{4} )</m></p></li>
    <li><p><image source="images/fig-ans-chap7-rev-57"  width="40%"><description>translation of reciprocal</description></image> </p></li>
</ol></p></answer>
</exercise>

<exercise number="58"><statement><p><m>y=\dfrac{2}{x^2-3x-10} </m></p></statement>
</exercise>

<exercise number="59"><statement><p><m>y=\dfrac{x-2}{x+3} </m></p></statement>
<answer><p><ol>
    <li><p>Horizontal asymptote <m>y = 1</m>; Vertical asymptote <m>x = -3</m>; <m>x</m>-intercept <m>(2,0)</m>; <m>y</m>-intercept <m>(0,\frac{-2}{3} )</m></p></li>
    <li><p><image source="images/fig-ans-chap7-rev-59"  width="40%"><description>translation of reciprocal</description></image> </p></li>
</ol></p></answer>
</exercise>

<exercise number="60"><statement><p><m>y=\dfrac{x-1}{x^2-2x-3} </m></p></statement>
</exercise>

<exercise number="61"><statement><p><m>y=\dfrac{3x^2}{x^2-4} </m></p></statement>
<answer><p><ol>
    <li><p>Horizontal asymptote <m>y = 3</m>; Vertical asymptote <m>x = \pm 2</m>; <m>x</m>-intercept <m>(0,0)</m>; <m>y</m>-intercept <m>(0,0 )</m></p></li>
    <li><p><image source="images/fig-ans-chap7-rev-61"  width="40%"><description>translation of reciprocal</description></image> </p></li>
</ol></p></answer>
</exercise>

<exercise number="62"><statement><p><m>y=\dfrac{2x^2-2}{x^2-9} </m></p></statement>
</exercise>
</exercisegroup>



<exercisegroup cols="2"><introduction><p>
    For Problems 63<ndash/>66,
    <ol label="a"><li><p>
        Use polynomial division to write the fraction in the form <m>y=\dfrac{k}{p(x)}+c </m>, where <m>k</m> and <m>c</m> are constants.
    </p></li>
    <li><p>Use transformations to sketch the graph.</p></li>
    </ol>
</p></introduction>

<exercise number="63"><statement><p><m>y=\dfrac{3x+4}{x+3} </m></p></statement>
<answer><p><ol cols="2">
    <li><p><m>y=\dfrac{-5}{x+3}+3 </m> </p></li>
    <li><p><image source="images/fig-ans-chap7-rev-63"  width="90%"><description>transformation of reciprocal</description></image> </p></li>
</ol></p></answer>
</exercise>

<exercise number="64"><statement><p><m>y=\dfrac{5x+1}{x-2} </m></p></statement>
</exercise>

<exercise number="65"><statement><p><m>y=\dfrac{x^2+2x+3}{(x+1)^2} </m></p></statement>
<answer><p><ol cols="2">
    <li><p><m>y=\dfrac{2}{(x+1)^2}+1 </m> </p></li>
    <li><p><image source="images/fig-ans-chap7-rev-65"  width="90%"><description>transformation of reciprocal-squared</description></image> </p></li>
</ol></p></answer>
</exercise>

<exercise number="66"><statement><p><m>y=\dfrac{x^2-4x+3}{(x-2)^2} </m></p></statement>
</exercise>
</exercisegroup>



<exercise number="67"><statement><p>
    The Explorer's Club is planning a canoe trip to travel <m>90</m> miles up the Lazy River and return in <m>4</m> days. Club members plan to paddle for <m>6</m> hours each day, and they know that the current in the Lazy River is <m>2</m> miles per hour.
    <ol>
        <li><p>Express the time it will take for the upstream journey as a function of their paddling speed in still water.</p></li>
        <li><p>Express the time it will take for the downstream journey as a function of their paddling speed in still water.</p></li>
        <li><p>Graph the sum of the two functions and find the point on the graph with <m>y</m>-coordinate <m>24</m>. Interpret the coordinates of the point in the context of the problem.</p></li>
        <li><p>The Explorer's Club would like to know what average paddling speed members must maintain in order to complete their trip in <m>4</m> days. Write an equation to describe this situation.</p></li>
        <li><p>Solve your equation to find the required paddling speed.</p></li>
    </ol>
</p></statement>
<answer><p><ol cols="2">
    <li><p><m>t_1=\dfrac{90}{v-2} </m></p></li>
    <li><p><m>t_2=\dfrac{90}{v+2} </m></p></li>
    <li><p><image source="images/fig-ans-chap7-rev-67.jpg"  width="95%"><description>curve</description></image> </p></li>
    <li><p><m>\dfrac{90}{v-2}+\dfrac{90}{v+2}=24 </m></p></li>
    <li><p><m>8</m> mph</p></li>
</ol></p></answer>
</exercise>

<exercise number="68"><statement><p>
    Pam lives on the banks of the Cedar River and makes frequent trips in her outboard motorboat. The boat travels at <m>20</m> miles per hour in still water.
    <ol>
        <li><p>Express the time it takes Pam to travel <m>8</m> miles upstream to the gas station as a function of the speed of the current.</p></li>
        <li><p>Express the time it takes Pam to travel <m>12</m> miles downstream to Marie's house as a function of the speed of the current.</p></li>
        <li><p>Graph the two functions in the same window, then find the coordinates of the intersection point. Interpret those coordinates in the context of the problem.</p></li>
        <li><p>Pam traveled to the gas station in the same time it took her to travel to Marie's house. Write an equation to describe this situation.</p></li>
        <li><p>Solve your equation to find the speed of the current in the Cedar River.</p></li>
    </ol>
</p></statement></exercise>

<exercise number="69"><statement><p>
    Mikala sells <m>\dfrac{320}{x} </m> bottles of bath oil per week if she charges <m>x</m> dollars per bottle. Her supplier can manufacture <m>\dfrac{1}{2}x+6</m> bottles per week if she sells it at <m>x</m> dollars per bottle.
    <ol>
        <li><p>Graph the demand function, <m>D(x) = \dfrac{320}{x}</m>, and the supply function, <m>S(x) = \dfrac{1}{2}x+6</m>, in the same window.</p></li>
        <li><p>Write and solve an equation to find the equilibrium price, that is, the price at which the supply equals the demand for bath oil. Label this point on your graph.</p></li>
    </ol>
</p></statement>
<answer><p><ol cols="2">
    <li><p><image source="images/fig-ans-chap7-rev-69.jpg"  width="90%"><description>supply and demand curves</description></image> </p></li>
    <li><p><m>\dfrac{320}{x}=\dfrac{1}{2}x+6</m>; <m>~\$20</m></p></li>
</ol></p></answer>
</exercise>

<exercise number="70"><statement><p>
    Tomoko sells <m>\dfrac{4800}{x} </m> exercise machines each month if the price of a machine is <m>x</m> dollars. On the other hand, her supplier can manufacture <m>2.5x+20</m> machines if she charges <m>x</m> dollars apiece for them.
    <ol>
        <li><p>Graph the demand function, <m>D(x) = \dfrac{4800}{x}</m>, and the supply function, <m>S(x) = 2.5x+20</m>, in the same window.</p></li>
        <li><p>Write and solve an equation to find the equilibrium price, that is, the price at which the supply equals the demand for exercise machines. Label this point on your graph.</p></li>
    </ol>
</p></statement>
</exercise>


<exercisegroup><introduction><p>Write and solve a proportion for each problem.</p></introduction>

<exercise number="71"><statement><p>A polling firm finds that <m>78</m> of the <m>300</m> randomly selected students at Citrus College play some musical instrument. Based on the poll, how many of the college’s <m>1150</m> students play a musical instrument?</p></statement>
<answer><p><m>299</m></p></answer>
</exercise>

<exercise number="72"><statement><p>Claire wants to make a scale model of Salem College. The largest building on campus, Lausanne Hall, is <m>60</m> feet tall, and her model of Lausanne Hall will be <m>8</m> inches tall. How tall should she make the model of Willamette Hall, which is <m>48</m> feet tall?</p></statement>
</exercise>
</exercisegroup>


<exercisegroup cols="2"><introdution><p>Solve.</p></introdution>

<exercise number="73"><statement><p><m>\dfrac{y+3}{y+5}=\dfrac{1}{3} </m></p></statement>
<answer><p><m>-2</m></p></answer>
</exercise>

<exercise number="74"><statement><p><m>\dfrac{z^2+2}{z^2-2}=3 </m></p></statement>
</exercise>

<exercise number="75"><statement><p><m>\dfrac{x}{x-2}=\dfrac{2}{x-2}+7 </m></p></statement>
<answer><p>No solution</p></answer>
</exercise>

<exercise number="76"><statement><p><m>\dfrac{3x}{x+1}-\dfrac{2}{x^2+x} =\dfrac{4}{x} </m></p></statement>
</exercise>

<exercise number="77"><statement><p><m>\dfrac{2}{a+1}+\dfrac{1}{a-1} =\dfrac{3a-1}{a^2-1} </m></p></statement>
<answer><p><m>a\ne -1, ~a\ne 1</m></p></answer>
</exercise>

<exercise number="78"><statement><p><m>\dfrac{2b-1}{b^2+2b}=\dfrac{4}{b+2} -\dfrac{1}{b} </m></p></statement>
</exercise>

<exercise number="79"><statement><p><m>\dfrac{-10}{u-2} = \dfrac{u-4}{u^2-u-2} + \dfrac{3}{u+1} </m></p></statement>
<answer><p><m>0</m></p></answer>
</exercise>

<exercise number="80"><statement><p><m>\dfrac{1}{t^2+t} + \dfrac{1}{t} = \dfrac{3}{t+1} </m></p></statement>
</exercise>
</exercisegroup>


<exercisegroup cols="2"><introduction><p>Solve for the indicated variable.</p></introduction>

<exercise number="81"><statement><p><m>V=C\left(1-\dfrac{t}{n} \right) </m>, for <m>n</m></p></statement>
<answer><p><m>n=\dfrac{Ct}{C-V} </m></p></answer>
</exercise>

<exercise number="82"><statement><p><m>r = \dfrac{dc}{1-ec} </m>, for <m>c</m></p></statement>
</exercise>

<exercise number="83"><statement><p><m>\dfrac{p}{q} = \dfrac{r}{q+r}  </m>, for <m>q</m></p></statement>
<answer><p><m>q=\dfrac{pr}{r-p} </m></p></answer>
</exercise>

<exercise number="84"><statement><p><m>I = \dfrac{E}{R+\dfrac{r}{n}} </m>, for <m>R</m></p></statement>
</exercise>

</exercisegroup>

</exercises>